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- Unit: Sampling distributions
- Populations, samples, and sampling distributions
- Populations, samples, and sampling distributions
- Sampling distributions and the bootstrap

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## Unit: Sampling distributions

Birth weights are recorded for all babies in a town. If we collect many random samples of 9 babies at a time, how do you think sample means will behave? Here again, we are working with a random variable, since random samples will have means that vary unpredictably in the short run but exhibit patterns in the long run. Based on our intuition and what we have learned about the behavior of sample proportions, we might expect the following about the distribution of sample means:. Center : Some sample means will be on the low side — say 3, grams or so — while others will be on the high side — say 4, grams or so.

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## Populations, samples, and sampling distributions

Understanding Regression Analysis pp Cite as. To this point, we have used regression analysis only to describe the relationship between two variables in a sample. However, in statistical analysis, we are not usually interested in the characteristics of a particular sample. More often, we are interested in estimating the characteristics of the population from which the sample was drawn. Whenever we wish to make statements about the characteristics of a population, based on the characteristics of a sample, we must rely on the logic of statistical inference.

When you want to determine information about a particular population characteristic for example, the mean , you usually take a random sample from that population because it is infeasible to measure the entire population. Using that sample, you calculate the corresponding sample characteristic, which is used to summarize information about the unknown population characteristic. The population characteristic of interest is called a parameter and the corresponding sample characteristic is the sample statistic or parameter estimate. Because the statistic is a summary of information about a parameter obtained from the sample, the value of a statistic depends on the particular sample that was drawn from the population. Its values change randomly from one random sample to the next one, therefore a statistic is a random quantity variable.

Sampling & Sampling Distributions: Basics. Peter Wludyka / samp1. 4. Theory for Sampling Distribution of the Mean. • For random samples of size n.

## Populations, samples, and sampling distributions

Analysing Economic Data pp Cite as. The issue of sampling from an underlying population is considered more formally, with the distinction being drawn between deductive and inductive statistical reasoning. To allow the ideas of statistical inference to be analysed, the concept of a simple random sample is introduced, along with the related ideas of accuracy and precision. The sampling distribution of the mean from a normal population is developed and the result extended, through the central limit theorem, to non-normal populations. The sampling distribution of the variance is then considered.

The sampling distribution of a statistic is the distribution of the statistic for all possible samples from the same population of a given size. Suppose you randomly sampled 10 women between the ages of 21 and 35 years from the population of women in Houston, Texas, and then computed the mean height of your sample. You would not expect your sample mean to be equal to the mean of all women in Houston. It might be somewhat lower or higher, but it would not equal the population mean exactly. Similarly, if you took a second sample of 10 women from the same population, you would not expect the mean of this second sample to equal the mean of the first sample.

### Sampling distributions and the bootstrap

In Example 6. The probability distribution is:. Whereas the distribution of the population is uniform, the sampling distribution of the mean has a shape approaching the shape of the familiar bell curve. This phenomenon of the sampling distribution of the mean taking on a bell shape even though the population distribution is not bell-shaped happens in general. Here is a somewhat more realistic example. The sampling distributions are:. What we are seeing in these examples does not depend on the particular population distributions involved.

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#### What Is a Sampling Distribution?

In statistics , a sampling distribution or finite-sample distribution is the probability distribution of a given random-sample -based statistic. If an arbitrarily large number of samples, each involving multiple observations data points , were separately used in order to compute one value of a statistic such as, for example, the sample mean or sample variance for each sample, then the sampling distribution is the probability distribution of the values that the statistic takes on. In many contexts, only one sample is observed, but the sampling distribution can be found theoretically. Sampling distributions are important in statistics because they provide a major simplification en route to statistical inference. More specifically, they allow analytical considerations to be based on the probability distribution of a statistic, rather than on the joint probability distribution of all the individual sample values. It may be considered as the distribution of the statistic for all possible samples from the same population of a given sample size.

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The sampling distribution of a statistic is the probability distribution of that statistic. Page 6. Sampling distribution of the sample mean. We take many random.